Finite Group Databasebeta

← C5×#74055 #14323138

Subgroups — C5×#74055 #14323138

Order 1920.

1080 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#67C211yesmaximal, characteristic
2C3×C5×#381C211yesmaximal, characteristic, complemented
3C5×#5212C211yesmaximal, characteristic, complemented
4C5×#4957C212yesmaximal
5C5×#4945C212yesmaximal, complemented
6C5×#4670C212yesmaximal, complemented
7C5×#4667C211yesmaximal, characteristic, complemented
8C5×#4664C211yesmaximal, characteristic, complemented
9C5×#4649C211yesmaximal, characteristic, complemented
10C5×#4648C211yesmaximal, characteristic, complemented
11C5×#4647C212yesmaximal, complemented
12C5×#2319—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C5×#217C2212yes—
18C2×C5×#217C2211yescharacteristic
19C4×C5×#67C418yescomplemented
20C2×C5×(C2×D3)⋊C4C2212yes—
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2211yescharacteristic
23C2×C5×C4⋊Dic3C2212yes—
24C2×C5×Dic3⋊C4C2212yes—
25C2×C5×Dic3⋊C4C2212yes—
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C4×C5×Dic3C2212yes—
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2211yescharacteristic
30C3×C5×#95C2212yes—
31C3×C5×#95C2211yescharacteristic
32C3×C5×#95C2211yescharacteristic
33C5×C3⋊#95C2212yes—
34C5×C3⋊#95C2211yescharacteristic
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2211yescharacteristic
37#74055C511yesmaximal, characteristic, direct product factor, Hall
38C2×C4×C5×D4—31no—
39C5×#390—32no—
40C5×#389—32no—
41C5×#388—32no—
42C5×#388—32no—
43C5×#381—31no—
44C5×#381D311yescharacteristic, complemented
45C5×#380—31no—
46C5×#380—31no—
47C4×C5×C4⋊C4—31no—
48C5×#376—31no—
49C24×C3×C5C2311yescharacteristic
50C23×C5×D3C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C2×C5×#67—24no—
58C2×C5×#67C2×C414yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C2×C4×C5×D3—42no—
65C2×C4×C5×D3C2×C414yes—
66C3×C5×C22⋊C4—42no—
67C5×#217—42no—
68C5×#217C2×C414yes—
69C5×(C2×D3)⋊C4—42no—
70C5×(C2×D3)⋊C4—42no—
71C5×(C2×D3)⋊C4C2×C414yes—
72C5×C4⋊Dic3—42no—
73C5×Dic3⋊C4—24no—
74C5×Dic3⋊C4—42no—
75C5×Dic3⋊C4C2×C414yes—
76C4×C5×Dic3—24no—
77C4×C5×Dic3—42no—
78C4×C5×Dic3C2×C414yes—
79C2×C4×#67C2×C511yescharacteristic
80C3×#381C2×C511yescharacteristic, complemented
81#5212C2×C511yescharacteristic, complemented
82#4957C2×C512yes—
83#4945C2×C512yescomplemented
84#4670C2×C512yescomplemented
85#4667C2×C511yescharacteristic, complemented
86#4664C2×C511yescharacteristic, complemented
87#4649C2×C511yescharacteristic, complemented
88#4648C2×C511yescharacteristic, complemented
89#4647C2×C512yescomplemented
90C22×C5×D4—31no—
91C23×C4×C5—31no—
92C23×C4×C5C2×D311yescharacteristic
93C4×C5×D4—38no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C4⋊C4—32no—
96C2×C5×C4⋊C4—32no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C22⋊C4—31no—
99C2×C5×C22⋊C4—31no—
100C2×C5×C22⋊C4—32no—
101C2×C5×C22⋊C4—32no—
102C2×C5×C22⋊C4—32no—
103C2×C5×C22⋊C4C2×D312yes—
104C2×C42×C5—31no—
105C2×C42×C5—32no—
106C2×C42×C5—32no—
107C5×#95—31no—
108C5×#95—32no—
109C5×#95—31no—
110C5×#95—32no—
111C5×#95C2×D311yescharacteristic
112C5×#95C2×D311yescharacteristic
113C5×#95C2×D312yes—
114#2319—31noSylow
115C23×C3×C5—22no—
116C23×C3×C5—21no—
117C23×C3×C5—22no—
118C23×C3×C5—21no—
119C23×C3×C5C22×C412yes—
120C23×C3×C5C2411yescharacteristic
121C22×C5×D3—22no—
122C22×C5×D3—21no—
123C22×C5×D3—22no—
124C22×C5×D3—21no—
125C22×C5×D3C22×C412yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C5×#67—28no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3C22×C412yes—
149C2×C5×Dic3C22×C412yes—
150C2×C5×Dic3#4114yes—
151C4×C5×D3—44no—
152C23×C3×C4C22×C511yescharacteristic
153C22×#67C22×C511yescharacteristic
154C22×C4×D3C22×C511yescharacteristic
155C2×C3×C22⋊C4C22×C512yes—
156C2×#217C22×C512yes—
157C2×#217C22×C511yescharacteristic
158C4×#67C4×C518yescomplemented
159C2×(C2×D3)⋊C4C22×C512yes—
160C2×(C2×D3)⋊C4C22×C512yes—
161C2×(C2×D3)⋊C4C22×C511yescharacteristic
162C2×C4⋊Dic3C22×C512yes—
163C2×Dic3⋊C4C22×C512yes—
164C2×Dic3⋊C4C22×C512yes—
165C2×Dic3⋊C4C22×C511yescharacteristic
166C2×C4×Dic3C22×C512yes—
167C2×C4×Dic3C22×C512yes—
168C2×C4×Dic3C22×C511yescharacteristic
169C3×#95C22×C512yes—
170C3×#95C22×C511yescharacteristic
171C3×#95C22×C511yescharacteristic
172C3⋊#95C22×C512yes—
173C3⋊#95C22×C511yescharacteristic
174C3⋊#95C22×C512yes—
175C3⋊#95C22×C511yescharacteristic
176C24×C5—31no—
177C24×C5C22×D311yescharacteristic
178C2×C5×D4—34no—
179C2×C5×D4—64no—
180C22×C4×C5—31no—
181C22×C4×C5—31no—
182C22×C4×C5—31no—
183C22×C4×C5—34no—
184C22×C4×C5—32no—
185C22×C4×C5—32no—
186C22×C4×C5—42no—
187C22×C4×C5—122no—
188C22×C4×C5C4×D314yes—
189C22×C4×C5C22×D312yes—
190C22×C4×C5C22×D312yes—
191C22×C4×C5C22×D311yescharacteristic
192C22×C4×C5C22×D311yescharacteristic
193C5×C4⋊C4—34no—
194C5×C4⋊C4—64no—
195C5×C4⋊C4—122no—
196C5×C4⋊C4—122no—
197C5×C22⋊C4—34no—
198C5×C22⋊C4—34no—
199C5×C22⋊C4—42no—
200C5×C22⋊C4—122no—
201C5×C22⋊C4—122no—
202C5×C22⋊C4—122no—
203C42×C5—34no—
204C42×C5—64no—
205C42×C5—122no—
206C2×C4×D4—31no—
207#390—32no—
208#389—32no—
209#388—32no—
210#388—32no—
211#381—31no—
212#381C5×D311yescharacteristic, complemented
213#380—31no—
214#380—31no—
215C4×C4⋊C4—31no—
216#376—31no—
217C22×C3×C5—22no—
218C22×C3×C5—22no—
219C22×C3×C5—22no—
220C22×C3×C5—42no—
221C22×C3×C5—42no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C2×#4112yes—
224C22×C3×C5H(5,𝔽2)12yes—
225C22×C3×C5H(5,𝔽2)11yescharacteristic
226C22×C3×C52−1+411yescharacteristic
227C2×C5×D3—22no—
228C2×C5×D3—22no—
229C2×C5×D3—22no—
230C2×C5×D3—42no—
231C2×C5×D3—42no—
232C3×C4×C5—22no—
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—42no—
236C5×Dic3—22no—
237C5×Dic3—22no—
238C5×Dic3—42no—
239C5×Dic3—42no—
240C5×Dic3#11714yes—
241C24×C3C23×C511yescharacteristic
242C23×D3C23×C511yescharacteristic
243C22×C3×C4—42no—
244C22×C3×C4C2×C4×C514yes—
245C22×C3×C4C23×C512yes—
246C22×C3×C4C23×C512yes—
247C22×C3×C4C23×C511yescharacteristic
248C22×C3×C4C23×C511yescharacteristic
249C2×#67—24no—
250C2×#67C2×C4×C514yes—
251C22×Dic3C23×C512yes—
252C22×Dic3C23×C512yes—
253C22×Dic3C23×C511yescharacteristic
254C22×Dic3C23×C511yescharacteristic
255C22×Dic3C23×C511yescharacteristic
256C2×C4×D3—42no—
257C2×C4×D3C2×C4×C514yes—
258C3×C22⋊C4—42no—
259#217—42no—
260#217C2×C4×C514yes—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4C2×C4×C514yes—
264C4⋊Dic3—42no—
265Dic3⋊C4—24no—
266Dic3⋊C4—42no—
267Dic3⋊C4C2×C4×C514yes—
268C4×Dic3—24no—
269C4×Dic3—42no—
270C4×Dic3C2×C4×C514yes—
271C23×C5—22no—
272C23×C5—22no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—32no—
276C23×C5—62no—
277C23×C5—62no—
278C23×C5—61no—
279C23×C5—61no—
280C23×C5C2×C4×D312yes—
281C23×C5C23×D311yescharacteristic
282C5×D4—68no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—32no—
292C2×C4×C5—34no—
293C2×C4×C5—32no—
294C2×C4×C5—32no—
295C2×C4×C5—44no—
296C2×C4×C5—62no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—62no—
300C2×C4×C5—62no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—124no—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5C2×C4×D312yes—
308C22×D4—31no—
309C23×C4—31no—
310C23×C4C2×C5×D311yescharacteristic
311C4×D4—38no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4C2×C5×D312yes—
322C2×C42—31no—
323C2×C42—32no—
324C2×C42—32no—
325#95—31no—
326#95—32no—
327#95—31no—
328#95—32no—
329#95C2×C5×D311yescharacteristic
330#95C2×C5×D311yescharacteristic
331#95C2×C5×D312yes—
332C2×C3×C5—42no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5#51811yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#54012yes—
337C2×C3×C5#54212yes—
338C5×D3—42no—
339C23×C3—22no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3—21no—
343C23×C3C22×C4×C512yes—
344C23×C3C24×C511yescharacteristic
345C22×D3—22no—
346C22×D3—21no—
347C22×D3—22no—
348C22×D3—21no—
349C22×D3C22×C4×C512yes—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—44no—
359C2×C3×C4C22×C4×C512yes—
360C2×C3×C4C22×C4×C512yes—
361#67—28no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3C22×C4×C512yes—
372C2×Dic3C22×C4×C512yes—
373C2×Dic3C22×C4×C512yes—
374C2×Dic3C5×#4114yes—
375C4×D3—44no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—22no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—62no—
384C22×C5—122no—
385C22×C5—122no—
386C22×C5C22×C4×D311yescharacteristic
387C22×C5#100412yes—
388C22×C5D3×#4112yes—
389C22×C5#100911yescharacteristic
390C22×C5#101011yescharacteristic
391C4×C5—22no—
392C4×C5—22no—
393C4×C5—34no—
394C4×C5—42no—
395C4×C5—42no—
396C4×C5—62no—
397C4×C5—62no—
398C4×C5—122no—
399C4×C5—122no—
400C24—31no—
401C24C22×C5×D311yescharacteristic
402C2×D4—34no—
403C2×D4—64no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—42no—
411C22×C4—122no—
412C22×C4C4×C5×D314yes—
413C22×C4C22×C5×D312yes—
414C22×C4C22×C5×D312yes—
415C22×C4C22×C5×D311yescharacteristic
416C22×C4C22×C5×D311yescharacteristic
417C4⋊C4—34no—
418C4⋊C4—64no—
419C4⋊C4—122no—
420C4⋊C4—122no—
421C22⋊C4—34no—
422C22⋊C4—34no—
423C22⋊C4—42no—
424C22⋊C4—122no—
425C22⋊C4—122no—
426C22⋊C4—122no—
427C42—34no—
428C42—64no—
429C42—122no—
430C3×C5#231911yescharacteristic, complemented, Hall
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3C23×C4×C511yescharacteristic
437C22×C3C2×C5×#4112yes—
438C22×C3C5×H(5,𝔽2)12yes—
439C22×C3C5×H(5,𝔽2)11yescharacteristic
440C22×C3C5×2−1+411yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—42no—
445C2×D3—42no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—42no—
450Dic3—22no—
451Dic3—22no—
452Dic3—42no—
453Dic3—42no—
454Dic3C5×#11714yes—
455C2×C5—42no—
456C2×C5—122no—
457C2×C5#548611yescharacteristic
458C2×C5D3×#11711yescharacteristic
459C2×C5#552311yescharacteristic
460C2×C5#566212yes—
461C2×C5#566312yes—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—32no—
467C23—62no—
468C23—62no—
469C23—61no—
470C23—61no—
471C23C2×C4×C5×D312yes—
472C23C23×C5×D311yescharacteristic
473D4—68no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—32no—
483C2×C4—34no—
484C2×C4—32no—
485C2×C4—32no—
486C2×C4—44no—
487C2×C4—62no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—124no—
497C2×C4C2×C4×C5×D312yes—
498C2×C4C2×C4×C5×D312yes—
499C2×C3—42no—
500C2×C3C2×C5×#11711yescharacteristic
501C2×C3C5×#51811yescharacteristic
502C2×C3C5×#52011yescharacteristic
503C2×C3C5×#54012yes—
504C2×C3C5×#54212yes—
505D3—42no—
506C5#7405511yescharacteristic, direct product factor, Sylow
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—62no—
513C22—62no—
514C22—62no—
515C22—122no—
516C22—122no—
517C22C22×C4×C5×D311yescharacteristic
518C22C5×#100412yes—
519C22C5×D3×#4112yes—
520C22C5×#100911yescharacteristic
521C22C5×#101011yescharacteristic
522C4—22no—
523C4—22no—
524C4—34no—
525C4—42no—
526C4—42no—
527C4—62no—
528C4—62no—
529C4—122no—
530C4—122no—
531C3C5×#231911yescharacteristic, complemented, Sylow
532C2—42no—
533C2—122no—
534C2C5×#548611yescharacteristic
535C2C5×D3×#11711yescharacteristic
536C2C5×#552311yescharacteristic
537C2C5×#566212yes—
538C2C5×#566312yes—
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#381C2
socleC23×C3×C5C24
prime radicalC22×C5×#67C22
cosocleC23×C3C24×C5
hypercenterC5×#381D3
hypocenterC3C5×#2319
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#381

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C5×#381C2semidirect
3. C5×#5212C2semidirect
5. C5×#4945C2semidirect
6. C5×#4670C2semidirect
7. C5×#4667C2semidirect
8. C5×#4664C2semidirect
9. C5×#4649C2semidirect
10. C5×#4648C2semidirect
11. C5×#4647C2semidirect
19. C4×C5×#67C4semidirect
37. #74055C5direct (N × H)
44. C5×#381D3semidirect
80. C3×#381C2×C5semidirect
81. #5212C2×C5semidirect
83. #4945C2×C5semidirect
84. #4670C2×C5semidirect
85. #4667C2×C5semidirect
86. #4664C2×C5semidirect
87. #4649C2×C5semidirect
88. #4648C2×C5semidirect
89. #4647C2×C5semidirect
158. C4×#67C4×C5semidirect
212. #381C5×D3semidirect
430. C3×C5#2319semidirect
506. C5#74055direct (N × H)
531. C3C5×#2319semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×C5×#672
C3×C5×#3812
C5×#52122
C5×#49572
C5×#49452
C5×#46702
C5×#46672
C5×#46642
C5×#46492
C5×#46482
C5×#46472
C5×#23193
#740555